Aspects of Thinness in Harmonic Analysis

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Harmonic analysts seek to develop mathematical theories that help to find solutions to problems that might have arisen from mathematical physics, electrical engineering and other branches of mathematics. ****My research in harmonic analysis is motivated, in part, by the harmonic analysis version of the Uncertainty Principle, a guiding philosophy loosely related to the Heisenberg Uncertainty Principle. Roughly speaking, it says information gained about a function has to be `paid for' by a corresponding loss of control on its Fourier transform, or vice versa. This is significant in applications. For instance, it implies that one cannot have a radio signal that is bounded in both time and frequencies. ****In my research, I develop qualitative and quantitative interpretations of this principle. I study functions, measures and sets that are thin or small, in some sense, and my goal is to understand the consequences of this. This topic has long been of interest to mathematicians for it is well known that `thin' objects are important in many problems and can exhibit interesting phenomena. For example, Weierstrass' famous example of a continuous, nowhere differentiable function was a trigonometric series with thin support of its transform, and the Cantor set, a thin but uncountable set, is important in many branches of mathematics. Recently, there have been important connections found relating thinness ideas from harmonic analysis with other areas of mathematics such as number theory and combinatorics.****My research program will have two major components. ****1. Thin sets: Sidon sets can be defined in terms of analytic properties of the functions whose Fourier transform is supported on the set. Although they are thin in an intuitive sense, their structure is complicated. The objective of this part of our program is to characterize Sidon sets in terms of thinner sets that are simpler to understand, a fundamental and long-standing problem in harmonic analysis. ****2. Thinly supported measures: **(a) The classical Cantor measure is a well-known example of a measure which has both small support and (somewhat) small Fourier transform. It is of interest throughout mathematics because of its pathological behavior and yet tractability. In the second part of the program we will develop techniques to quantify the local behavior of Cantor-like measures and other measures that arise from an iterative construction, but have overlap. These are important, current topics in fractal geometry.****(b) Lie groups are often used in physics as they describe real-world geometry. Orbital measures are elementary components in this setting and like Cantor measures have small support and small transform. The final part of my research program is to understand the smoothness properties of this class of thin measures and their associated geometric structures. I anticipate this will have important applications for harmonic analysis on Lie groups.**